Anarki doesn't have mailgun support yet. It will reset passwords if your server supports sending email using the local unix sendmail command. So I can think of two options:
a) Does mailgun provide a sendmail command that you can download? If so, just installing that might be all you need.
b) Anarki has rudimentary support for GET and POST on arbitrary urls.
Can you try to figure out the right URL to put into the get-url call or similar post-url call? They're implemented in lib/web.arc, and feel free to ask us questions to help get you on your way.